Ultrametric spaces generated by labeled star graphs

Oleksiy Dovgoshey; Olga Rovenska

Ultrametric spaces generated by labeled star graphs

T0 review · 0 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that an ultrametric space is generated by a labeled star graph exactly when some point's distances never exceed rival distances, and that the metric isometry group equals the graph automorphism group exactly when the…

desk verdict Clean, correct structural results for star-generated ultrametric spaces; the proof typos are minor and repair without changing the theorems. read the letter →

arxiv 2502.01260 v1 pith:LBTJ7VTD submitted 2025-02-03 math.GN

Oleksiy Dovgoshey , Olga Rovenska This is my paper · ORCID

classification math.GN MSC 54E35

keywords ultrametricspaceslabeledstargraphsgraphisomorphismisometryofmetricdistancespectrumcenterpointgroup

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Ultrametric spaces are metric spaces whose distances obey the strong triangle inequality $d(x,y)\le\max\{d(x,z),d(z,y)\}$; they arise in hierarchical classification and in number theory. The paper characterizes the subclass generated by a labeled star graph: such a space must contain a point $x_0$ with $d(x_0,x)\le d(y,x)$ for every pair of distinct points $x,y$ different from $x_0$. When this holds, a canonical labeled star graph is obtained by labeling the center $0$ and every other vertex by its distance from $x_0$. The paper further proves that the generating star graph is unique up to isomorphism exactly when the set of nonzero distances has infimum $0$, and that the self-isometry group of the space equals the self-isomorphism group of every generating star graph exactly when that distance set has no least element. The upshot is a purely metric test for when a star-shaped labeled graph is the right combinatorial model for an ultrametric space.

What carries the argument

The machinery is the labeled star graph $S(l)$—one center adjacent to every other vertex—together with the distance formula $d_l(u,v)=\max_{w\in V(P)}l(w)$, where $P$ is the unique path joining $u$ and $v$; for a star this reduces to a maximum over endpoint labels. The load-bearing criterion is condition (2.1), the existence of a point $x_0$ whose distance to any other point never exceeds any other distance involving that point. The remaining mechanism is order-theoretic: the set $D_0$ of nonzero distances, through whether it has infimum $0$ and whether it has a least element, controls uniqueness of the generating star graph and the equality of isometry and graph-automorphism groups.

What would settle it

Take a four-point ultrametric space with no point $x_0$ satisfying $d(x_0,x)\le d(y,x)$ for all distinct $x,y\ne x_0$ (the spaces in Figure 3 are examples) and enumerate every labeled star graph on four vertices, checking that none reproduces the distance table; Theorem 2.1 would fail if one did. For Theorem 3.2, try to construct a star-generable space whose positive distance set has no least element yet admits a distance-preserving map that moves the center; finding one would disprove the group equality.

Watch

Extended reading notes

Core claim

The central discovery is a metric certificate for star generation. Theorem 2.1 states that an ultrametric space $(X,d)$ belongs to the class $\mathbf{US}$—there is a labeled star graph $S(l)$ with $X=V(S)$ and $d=d_l$—if and only if some point $x_0\in X$ satisfies $d(x_0,x)\le d(y,x)$ whenever $x_0\ne x\ne y$. The point $x_0$ acts as a center: every distance from it to another point is no larger than any rival distance to that point from a third point. The paper then proves that the star representation is rigid exactly when the positive distances accumulate at $0$, i.e. $\inf D_0(X)=0$, and that the equality $\operatorname{Iso}(X,d)=\operatorname{Iso}S(l)$ holds for every generating labeled star graph exactly when the set $D_0$ of nonzero distances has no least element. The least-element condition is what prevents a minimally labeled leaf from being swapped with the center to create a metric symmetry invisible to the graph.

Load-bearing premise

The load-bearing premise is that the two results quoted from reference [15]—that the vertex-label maximum rule defines an ultrametric exactly when no edge has two zero labels, and that isomorphisms of labeled trees preserve the generated distances—are valid for infinite trees; the paper uses both as black boxes without re-proving them.

Editorial extensions

If this is right

Every ultrametric space with at most three points is star-generable, since every ultrametric triangle is isosceles with base no larger than the legs (Corollary 2.1).
The space $(\mathbb{R}^+,d_+)$ with $d_+(p,q)=\max\{p,q\}$ for $p\ne q$ is a US-space with a unique generating labeled star graph, because its positive distances have infimum $0$ (Examples 2.1 and 2.2).
Whenever a star-generable space has positive distances accumulating at $0$, every self-isometry of the space is a self-isomorphism of every labeled star graph generating it (Corollary 3.2).
If the positive distance set has no least element, the equality of the isometry group and the graph-automorphism group still holds; if it has a least element, swapping the center with a minimally labeled leaf produces a self-isometry that is not a graph self-isomorphism (Theorem 3.2).
Four-point ultrametric spaces need not be star-generable: the two spaces displayed in Figure 3 fail the center-point test and are not in $\mathbf{US}$ (Example 4.1).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct algorithmic reading not stated in the paper: condition (2.1) can be checked in quadratic time by precomputing, for each point $x$, its minimal distance to any other point, and then testing whether a candidate $x_0$ stays at or below that minimum for every $x$.
The construction in Theorem 2.1 is effectively canonical: labels are just distances to the distinguished point, so any finite ultrametric dataset passing the test yields a star model whose vertex labels are measured distances, not free parameters.
The no-least-element dichotomy implies a practical caveat: in a finite star-generable space, a unique smallest positive distance is exactly what creates metric symmetries that ignore the graph, so graph-based symmetry counts should be trusted only when no such minimum exists.
The compactness conjecture points to a testable boundary: an infinite star-generated space should be compact exactly when its leaf labels can be arranged as a sequence decreasing to $0$; checking whether the metric alone forces such an ordering would clarify the transition from star models to ray models.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

Quick take: this is a clean, correct little paper that characterizes ultrametric spaces generated by labeled star graphs. Theorem 2.1 gives a simple metric test: existence of a point x0 whose distance to any other point is no larger than any competing distance involving that point. Theorem 3.2 gives a sharp condition for equality of the isometry group and the labeled-star automorphism group: the set of nonzero distances has no least element. The proofs are mostly direct and the math holds up.

What is new: the class US is a special case of the labeled-tree framework from Dovgoshey's earlier paper [15], but the star-specific metric characterization and the group-equality criterion are new, not mere restatements. The examples are helpful and the two conjectures in Section 4 are clearly marked.

The soft spots are minor. The reader flagged two typos: inequality (2.33) is asserted for every n but the left side is 0 if x_n equals the center, and equation (3.34) writes D0(X,d*) = 0 instead of inf D0(X,d*) = 0. Both are repairable and do not affect the theorems. The paper imports two lemmas from [15]; they are elementary and the stress-test note confirms they apply to infinite star graphs without issue. Self-citation is present but not circular; the quoted lemmas are not the target results.

My verdict: this deserves a serious referee. The paper is sound, the statements are crisp, and the topic, while narrow, is a natural piece of the labeled-tree program. It is not a blockbuster but it is a solid contribution. With these typos fixed and the proof of (3.34) made explicit, I would accept after minor revision.

Recommendation: engage with it. Send it to a competent referee. I'd bring it to a reading group on ultrametrics, though not to a general topology seminar.

Referee Report

0 major / 5 minor

Summary. The paper studies the class US of ultrametric spaces that arise as (V(S), d_l) for a star graph S with a nonnegative, non-degenerate vertex labeling l, where d_l(u,v) is the maximum label on the unique path between u and v. Theorem 2.1 gives a metric characterization: an ultrametric space (X,d) belongs to US iff there is a point x0 such that d(x0,x) ≤ d(y,x) whenever x0 ≠ x ≠ y. Theorem 2.2 characterizes, for US-spaces, when the generating labeled star graph is unique up to isomorphism: this happens exactly when inf D0(X) = 0. In Section 3, Theorem 3.1 relates self-isometries of the ultrametric space to isomorphisms of the generating labeled trees, and Theorem 3.2 shows that Iso(X,d) = Iso S(l) for every generating labeled star graph S(l) iff the set D0(X,d) has no least element. The paper closes with examples and two conjectures concerning finite four-point obstructions and compactness of infinite US-spaces.

Significance. The results are clean and checkable: Theorem 2.1 is a genuinely simple metric criterion, and Theorem 3.2 gives a complete answer to a natural rigidity question for star-generated ultrametric spaces. The proofs are constructive and there are no fitted parameters or ad hoc assumptions. The paper depends on two lemmas imported from the first author's earlier paper [15], namely Theorem 1.1 and Proposition 3.1; I checked their application to possibly infinite star graphs and found them elementary and correct, so I do not see a circularity or correctness risk. The contribution is modest and incremental, but it is solid and should be of interest to researchers working on ultrametrics and labeled trees.

minor comments (5)

[§2, Theorem 2.2 proof] Inequality (2.33) is asserted for every n, but its left side is 0 when x_n = c1; the argument should split into the cases x_n = c1 and x_n ≠ c1, applying the inequality on a subsequence or switching to y_n if necessary. The adjacent use of (2.3) to derive d(c_i,x_n) ≤ d(x_n,y_n) needs the same caveat for x_n = c_i, where the inequality is trivial rather than an instance of (2.3).
[§3, Theorem 3.2 proof, Eq. (3.34)] The displayed equality D0(X,d*) = 0 should be inf D0(X,d*) = 0, since after subtracting a non-attained infimum the value 0 is not actually a distance; this is exactly the hypothesis needed for Corollary 3.2, so the correction does not affect the rest of the proof.
[§3, Theorem 3.1 proof, (ii)⇒(i)] The proof says 'Let T2(l2) be a labeled star graph generating (X,d)', although the theorem is stated for arbitrary labeled trees; the same argument works for labeled trees, so the wording should be corrected.
[§1, imported lemmas] Theorem 1.1 and Proposition 3.1 are quoted from [15] without proof; both are short and their extension to infinite star graphs is immediate, but for self-containedness the authors should either include proofs or restate the needed results explicitly.
[§2, Theorem 2.2 statement] Statement (ii) is worded as 'S1(l1) are isomorphic to S2(l2)'; it should read 'S1(l1) and S2(l2) are isomorphic', and similarly in the proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; central proofs are constructive and imported lemmas are independent.

full rationale

The derivation chain is self-contained for the paper's central claims. Theorem 2.1 proves both directions directly: (i)->(ii) reads off the center inequality from the definition d_l(c,x)=max{l(c),l(x)} <= max{l(y),l(c),l(x)} for a star, and (ii)->(i) constructs the labeling l(x)=0 at x0 and l(x)=d(x0,x) otherwise, then verifies d=d_l using the strong triangle inequality and the defining inequality (2.1); no quantity is fitted from the data it is asked to predict. Theorem 2.2 likewise proves uniqueness from inf D0 = 0 by direct limit arguments and the identity d(c,x)=l(x) when l(c)=0. The remaining group theorems rely on Theorem 1.1 and Proposition 3.1, imported from [15] by the first author, but these are elementary, parameter-free lemmas whose stated assumptions concern labeled-tree isomorphism and non-degenerate labelings, not the article's target claims. They are used as independent support, not as conclusions smuggled into premises. The typo in (3.34) (D0(X,d*)=0 instead of inf D0(X,d*)=0) is repairable and does not make the argument circular. There are no fitted parameters, no prediction of a subset from which a parameter was fit, and no uniqueness assertion imported solely from authors' prior authority. Score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on two quoted lemmas from the first author's earlier paper [15], the non-degeneracy criterion and the isomorphism-implies-isometry lemma, plus standard ultrametric facts. No new entities or fitted parameters are introduced.

assumptions (4)

domain assumption Theorem 1.1: d_l is an ultrametric on V(T) iff max{l(u), l(v)} > 0 for every edge {u,v} (non-degenerate labeling).
Quoted as a direct corollary of Proposition 3.2 from [15]. The whole paper restricts to non-degenerate labelings so that d_l is an ultrametric; if this criterion failed, the class US could contain non-metric functions.
domain assumption Proposition 3.1: an isomorphism f of labeled trees T1(l1) and T2(l2) yields d_l1(u,v)=d_l2(f(u),f(v)) for all u,v.
Stated as Lemma 3.11 from [15] and used in the proof of Theorem 3.1 and hence in Theorem 3.2 and its corollaries. It is not re-proved or formally verified here.
standard math Standard ultrametric facts: every triangle is isosceles with base no larger than the equal sides, and convergent sequences have unique limits.
Used in Corollary 2.1 and in Theorem 2.2 for limits of sequences and equality of centers.
standard math The infimum of the empty set in [0,+infinity] is +infinity.
Used in Theorem 2.2 for the singleton case to show inf D0 != 0 and hence no unique generator.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Ultrametric spaces generated by labeled star graphs." pith.science (2026). https://pith.science/paper/LBTJ7VTD

@misc{pith2026250201260,
  author       = {Pith},
  title        = {Pith review of: Ultrametric spaces generated by labeled star graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LBTJ7VTD}},
  note         = {Machine review of arXiv:2502.01260}
}

abstract

For arbitrary star graph $S$ with a non-degenerate vertex labeling $l\colon V(S) \to \mathbb{R}^+$ we denote by $d_l$ the corresponding ultrametric on the vertex set $V(S)$ of $S$. We characterize the class $\bf US$ of all ultrametric spaces $(V(S), d_l)$ up to isometry. We also find the necessary and sufficient conditions under which the group of all self-isometries of ultrametric space $(V(S), d_l)$ coincides with the group of all self-isomorphisms of the labeled star graph $S(l)$.

Figures

Figures reproduced from arXiv: 2502.01260 by the authors.

**Figure 2.** The labeled path T1(l1) and the labeled star graph T2(l2) generate isometric spaces (V (T1), dl1 ) and (V (T2), dl2 ) if 0 < a1 ≤ a2 ≤ a3 ≤ a4. The next Proposition 3.2 shows that the isometry of US-spaces implies that the star graphs generated these spaces are isomorphic as free trees. Proposition 3.2. Let (X1, d1) and (X2, d2) be isometric US-spaces and let labeled star-graphs S1(l1) and S2(l2) generate (X1, d1) … view at source ↗

**Figure 3.** (X4, d4) and (Y4, ρ4) are not US-spaces. Example 4.1. The four-point ultrametric spaces (X4, d4) and (Y4, ρ4) depicted in [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: (V (S), dl ∗ ) ∈ US is a completion of (V (R), dl) ∈/ US. Conjecture 4.2. Let (X, d) be an infinite US-space generated by labeled star graph with a center c, let X0 := X \ {c} and let d0 be the restriction of d on the set X0 × X0. Then the following statements are equi…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Totally bounded ultrametric spaces and locally finite trees
math.GN 2025-02 accept novelty 7.0 of 10

Totally bounded ultrametric spaces are represented, up to isometry of their completions, by locally finite labeled trees whose vertices are open balls labeled by their diameters.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · cited by 1 Pith paper

[15]

Dovgoshey, Isomorphism of trees and isometry of ultrametric spaces, Theory and Applications of Graphs, 7 (2020), no

O. Dovgoshey, Isomorphism of trees and isometry of ultrametric spaces, Theory and Applications of Graphs, 7 (2020), no. 2, Art. 3

work page 2020
[1]

Bilet, O

V. Bilet, O. Dovgoshey, and Yu. Kononov, Ultrametrics and Com- plete Multipartite Graphs , Theory and Applications of Graphs, 9 (2022), no. 1, Article 8

work page 2022
[2]

Bruhn, and R

H. Bruhn, and R. Diestel, Duality in inﬁnite graphs , Combinatorics, Probability and Computing, 15 (2006), 75–90

work page 2006
[3]

Bruhn, R

H. Bruhn, R. Diestel, and M. Stein, Cycle-cocycle partitions and faithful cycle covers for locally ﬁnite graphs , Journal of Graph The- ory, 50 (2005), 150–161

work page 2005
[4]

Bruhn, and M

H. Bruhn, and M. Stein, Duality of ends , Combinatorics, Probabil- ity and Computing, 19 (2010), no. 1, 47–60

work page 2010
[5]

Delhomm´ e, C

C. Delhomm´ e, C. Laﬂamme, M. Pouzet, and N. Sauer, Indivisi- ble ultrametric spaces , Topology and its Applications, 155 (2008), no. 14, 1462–1478. 20

work page 2008
[6]

Diestel, End spaces and spanning trees, Journal of Combinatorial Theory, Series B, 96 (2006), no

R. Diestel, End spaces and spanning trees, Journal of Combinatorial Theory, Series B, 96 (2006), no. 6, 846–854

work page 2006
[7]

Diestel, Locally ﬁnite graphs with ends: A topological approach, I

R. Diestel, Locally ﬁnite graphs with ends: A topological approach, I. Basic theory , Discrete Mathematics, 311 (2011), no. 15, 1423– 1447

work page 2011

Show all 31 references

[8]

Diestel, Ends and Tangles , Abhandlungen aus dem Mathematis- chen Seminar der Universit¨ at Hamburg 87 (2017), 223–244

R. Diestel, Ends and Tangles , Abhandlungen aus dem Mathematis- chen Seminar der Universit¨ at Hamburg 87 (2017), 223–244

2017
[9]

Diestel, Graph Theory, ﬁfth ed., Graduate Texts in Mathematics, 173, Springer, Berlin, 2017

R. Diestel, Graph Theory, ﬁfth ed., Graduate Texts in Mathematics, 173, Springer, Berlin, 2017

2017
[10]

Diestel, and D

R. Diestel, and D. K¨ uhn, Topological paths, cycles and spanning trees in inﬁnite graphs , European Journal of Combinatorics, 25 (2004), 835–862

2004
[11]

Diestel, and J

R. Diestel, and J. Pott, Dual trees must share their ends , Journal of Combinatorial Theory, Series B, 123 (2017), 32–53

2017
[12]

Diestel, and P

R. Diestel, and P. Spr¨ ussel, The fundamental group of a locally ﬁnite graph with ends , Advances in Mathematics, 226 (2011), no. 3, 2643–2675

2011
[13]

Diestel, and P

R. Diestel, and P. Spr¨ ussel, On the homology of locally compact spaces with ends, Topology and its Applications, 158 (2011), no. 13, 1626–1639

2011
[14]

Diestel, and P

R. Diestel, and P. Spr¨ ussel, Locally ﬁnite graphs with ends: A topo- logical approach, III. Fundamental group and homology , Discrete Mathematics, 312 (2012), no. 1, 21–29

2012
[16]

Dovgoshey, and A

O. Dovgoshey, and A. Kostikov, Delhomme–Laﬂamme–Pouzet– Sauer space as groupoid , Journal of Mathematical Sciences, 284 (2024), no. 3, 315–328

2024
[17]

Dovgoshey, and A

O. Dovgoshey, and A. Kostikov, Locally ﬁnite ultrametric spaces and labeled trees , Journal of Mathematical Sciences, 276 (2023), no. 5, 614–637

2023
[18]

Dovgoshey, and M

O. Dovgoshey, and M. K¨ u¸ c¨ ukaslan,Labeled trees generating com- plete, compact, and discrete ultrametric spaces , Annals of Combi- natorics, 26 (2022), 613–642. 21

2022
[19]

Dovgoshey, O

O. Dovgoshey, O. Martio, and M. Vuorinen, Metrization of weighted graphs, Annals of Combinatorics, 17 (2013), 455–476

2013
[20]

Dovgoshey, and E

O. Dovgoshey, and E. Petrov, Subdominant pseudoultrametric on graphs, Sbornik: Mathematics, 204 (2013), no. 8, 1131–1151

2013
[21]

Dovgoshey, and E

O. Dovgoshey, and E. Petrov, On some extremal properties of ﬁ- nite ultrametric spaces , p-Adic Numbers, Ultrametric Analysis and Applications, 12 (2020), no. 1, 1–11

2020
[22]

Dovgoshey, and E

O. Dovgoshey, and E. Petrov, Weak similarities of metric and semi- metric spaces, Acta Mathematica Hungarica, 141 (2013), 301–319

2013
[23]

Dovgoshey, E

O. Dovgoshey, E. Petrov, and H.-M. Teichert, On spaces extremal for the Gomory-Hu inequality , p-Adic Numbers, Ultrametric Anal- ysis and Applications, 7 (2015), no. 2, 133–142

2015
[24]

Dovgoshey, E

O. Dovgoshey, E. Petrov, and H.-M. Teichert, How rigid the ﬁnite ultrametric spaces can be? Journal of Fixed Point Theory and Ap- plications, 19 (2017), no. 2, 1083–1102

2017
[25]

Dovgoshey, and V

O. Dovgoshey, and V. Shcherbak, The range of ultrametrics, com- pactness, and separability , Topology and its Applications, 305 (2022), 107899

2022
[26]

Dovgoshey, and V

O. Dovgoshey, and V. Vito, Totally bounded ultrametric spaces gen- erated by labeled rays , arXiv:2402.15774, 2024

2024 arXiv
[27]

Gurvich, and M

V. Gurvich, and M. Vyalyi, Characterizing (quasi-)ultrametric ﬁ- nite spaces in terms of (directed) graphs , Discrete Applied Mathe- matics, 160 (2012), no. 12, 1742–1756

2012
[28]

Ishiki, Constructions of Urysohn universal ultrametric spaces , p- Adic Numbers, Ultrametric Analysis and Applications, 15, ( 2023), no

Y. Ishiki, Constructions of Urysohn universal ultrametric spaces , p- Adic Numbers, Ultrametric Analysis and Applications, 15, ( 2023), no. 4, 266–283

2023
[29]

A. J. Lemin, On Gelgfand’s problem concerning graphs, lattices, and ultrametric spaces , AAA62 Workshop on General Algebra –
[30]

Petrov, and A

E. Petrov, and A. Dovgoshey, On the Gomory-Hu inequality , Jour- nal of Mathematical Sciences, 198 (2014), no. 4, 392–411. CONTACT INFORMATION 22 Oleksiy Dovgoshey Institute of Applied Mathematics and Mechanics of NASU, Slo vyansk, Ukraine, Department of Mathematics and Statis...

2014
[62]

Arbeitstagung Allgemeine Algebra (Linz, Austria), Jun e 2001, pp. 12–13

2001

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [15]

Dovgoshey, Isomorphism of trees and isometry of ultrametric spaces, Theory and Applications of Graphs, 7 (2020), no

O. Dovgoshey, Isomorphism of trees and isometry of ultrametric spaces, Theory and Applications of Graphs, 7 (2020), no. 2, Art. 3

work page 2020

[2] [1]

Bilet, O

V. Bilet, O. Dovgoshey, and Yu. Kononov, Ultrametrics and Com- plete Multipartite Graphs , Theory and Applications of Graphs, 9 (2022), no. 1, Article 8

work page 2022

[3] [2]

Bruhn, and R

H. Bruhn, and R. Diestel, Duality in inﬁnite graphs , Combinatorics, Probability and Computing, 15 (2006), 75–90

work page 2006

[4] [3]

Bruhn, R

H. Bruhn, R. Diestel, and M. Stein, Cycle-cocycle partitions and faithful cycle covers for locally ﬁnite graphs , Journal of Graph The- ory, 50 (2005), 150–161

work page 2005

[5] [4]

Bruhn, and M

H. Bruhn, and M. Stein, Duality of ends , Combinatorics, Probabil- ity and Computing, 19 (2010), no. 1, 47–60

work page 2010

[6] [5]

Delhomm´ e, C

C. Delhomm´ e, C. Laﬂamme, M. Pouzet, and N. Sauer, Indivisi- ble ultrametric spaces , Topology and its Applications, 155 (2008), no. 14, 1462–1478. 20

work page 2008

[7] [6]

Diestel, End spaces and spanning trees, Journal of Combinatorial Theory, Series B, 96 (2006), no

R. Diestel, End spaces and spanning trees, Journal of Combinatorial Theory, Series B, 96 (2006), no. 6, 846–854

work page 2006

[8] [7]

Diestel, Locally ﬁnite graphs with ends: A topological approach, I

R. Diestel, Locally ﬁnite graphs with ends: A topological approach, I. Basic theory , Discrete Mathematics, 311 (2011), no. 15, 1423– 1447

work page 2011